By using the particle probability density we analyze the spin spin phase fluctuations because of stochastic particle migraecho attenuation of particles, diffusing in a bounded region. It tion in a nonuniform magnetic field provides a means to expand a nonuniform spin phase distribution into a series of waves that characterize the geometry and boundary
conditions of confinement. Random motion disrupts the initial phase structure created by applied gradients and consequently discords its structure waves. By assuming the spin phase fluctuawhere the phase term is given by tion and/or the randomness of spin phase distribution in the subensemble as a Gaussian stochastic process, we derive a new analytical expression for the echo attenuation related to the particle veloc-
ity correlation. For a diffusion in porous structure we get the expression featuring the same ''diffusive diffraction'' patterns as those being found and explained by P. T. Callaghan and A. Coy with G i being the gradient of the magnetic field at the ith (''Principles of Nuclear Magnetic Resonance Microscopy,'' Oxford spin site. In the special case when two gradient pulses of Univ. Press, Oxford (1991); J. Chem. Phys. 101, 4599-4609 (1994)) duration d separated by time D are very narrow, the average with the use of propagator theory. With the new approach we cast »rrr… L , Eq. [1], can be evaluated by using the diffusion a new light on the phenomena and derive analitically how the propagators (4). Denoting the positions of a particle at time diffusive diffractions appear when the sequence of finite or even 0 and D as r and r, it gives the normalized echo amplitude modulated gradients are applied. The method takes into account the non-Markovian character of restricted diffusion, and therefore E(q, D) the echo dependence on the diffusion lengths and on the strength of applied gradient differs from the results of authors assuming
Å ͐ dr◊(r) ͐ drP(r, DÉr, 0)e iqr(r=0r) , [3] the Markovian diffusion either by dealing with the diffusion propagators or by the computer simulation of Fick's diffusion. ᭧ 1998
Academic Press
where q Å gGd. This method has been successfully used for a restricted diffusion with the infinitely short gradient pulses (SGP approximation). However, it does not provide 339